Transcript Unit 1: Equations and Functions

UNIT 1: EQUATIONS AND FUNCTIONS
VARIABLE EXPRESSIONS
VOCABULARY QUEST

Directions:

Look up each of the
following vocabulary
words on Google.



Give a definition and an
example
Once you are done,
compare your definitions
with two other students in
the room
Once everyone is done,
we will compare to
everyone’s definitions

Vocabulary:
 Algebra
 Generalize
 Variables
 Variable Expression
 Equation
 Real
 Evaluate
VOCABULARY DEFINITIONS
Algebra: The part of mathematics in which letters
and other symbols are used to represent numbers
and quantities in formulae and equations
 Generalize: To infer from previous conclusions.
 Variables: An element, feature, or factor that is
liable to vary or change or a quantity that during a
calculation is assumed to vary or be capable of
varying in value.
 Variable Expression: A combination of numbers
(or constants), operations, and variables to create
a mathematical expression.

VOCABULARY DEFINITIONS
Equation: A statement that the values of two
mathematical expressions are equal (indicated by
the sign =).
 Real: Actually existing as a thing or occurring in
fact
 Evaluate: Find a numerical expression or
equivalent for (an equation, formula, or function)

MATHEMATICAL OPERATIONS
 Add
(+) :uses sum of, increase of, more
than
 Subtract ( - ) :uses difference of, less than,
fewer
 Multiply ( x or ● ) :uses product of, double,
triple, etc
 Divide ( ÷ ) :uses quotient of, divided by, per
 Square ( ² )
 Square Root ( √ )
MATHEMATICAL VARIABLES

Letters which take the place of numbers until
solved.
 Examples:
X
Y
Z
A
B
Any letter you wish the variable to be!!!!
EXAMPLE 1
 Write
an algebraic expression for the
perimeter and area of a rectangle.
L
 Perimeter = ?
 Area = ?
W
VARIABLE EXPRESSION
VS VARIABLE EQUATION
 An
example of a variable expression from
the previous example is 2w + 2l
 An
example of a variable equation from the
previous example is P = 2w + 2l
 An
expression does not have an equal sign
where an equation has an equal sign.
EVALUATE ALGEBRAIC EXPRESSIONS
 Find
the value of 2x + 7
When x = 12
When
x=1
When
x = 10
EVALUATE ALGEBRAIC EXPRESSIONS
 Find
the value of -9x + 2
When x = -1
When
x=1
When
x=0
EVALUATE ALGEBRAIC EXPRESSIONS
WITH MORE THAN ONE VARIABLE

Many expressions have more than one variable.
 To solve, substitute the appropriate value into
the appropriate place
 Example:
P
Perimeter and Area of a rectangle
= 2l + 2w
A=lxw
Find the perimeter and area of a rectangle
with:
Length of 12 and a width of 10
EXPRESSIONS WITH EXPONENTS
 Exponents
are short hand notations for
repeated multiplication
 2⁴
= 2*2*2*2
The base is 2 and the exponent is 4.
Means that the base is multiplied by itself
4 times.
EXPRESSIONS WITH EXPONENTS
 Can
also write variables with exponents
 x³
= x*x*x
The base is x and the exponent is 3
Means that the base is multiplied by itself 3
times
 When
x is negative and the power is even
the answer will be positive.
 When x is negative and the power is odd
the answer will be negative.
 Why is this?
EXPRESSIONS WITH EXPONENTS
 Find
the value of 5x² - 4y
When x = -4 and y = 5
EXPRESSIONS WITH EXPONENTS
 Find
the value of 2x² - 3x³ + 5
When x = -5
EXPRESSIONS WITH EXPONENTS
 Find
the value of
 When
𝑥2𝑦2
𝑥 2 +𝑦 2
x = 2 and y = -4
ACTIVITY

Each of you will pick out a random card. One
will be an expression and one will be an
answer. You are to find your partner and sit
together, waiting for more instructions.
ACTIVITY
ORDER OF OPERATIONS
EVALUATE THE EXPRESSION

2 + 4 x 7 – 1 = ???
How did different groups interpret this
problem?
 What is the correct way?

ORDER OF OPERATIONS
PEMDAS
 P- Parentheses
 E- Exponents
 M- Multiplication
 D- Division
 A- Addition
 S- Subtraction

With multiplication/ division as
well as addition/ subtraction, you
work from left to right completing
BOTH as they arise in the
problem. They are
interchangeable.
EXAMPLES

4 – 7 – 11 – 2

4 – (7 – 11) – 2

4 – [ 7 – (11 – 2)]
EXAMPLES

3x5–7÷2

3 x (5 – 7) ÷ 2

(3 x 5) – (7 ÷ 2)
EXAMPLES

2 – (19 – 7)² x (4³ - 2)
ORDER OF OPERATIONS BINGO

Directions:
You are going to receive a BINGO sheet and a sheet of
expressions.
 Cut out the expressions, solve them and put them in
the correct column.

 The
correct column means if the solution is between 1 and 10,
the expression goes in the ‘B’ column, etc.
Tape 5 expressions to the BINGO board in each column.
 We are going to play BINGO; the solutions to the
expressions will be called out. If the solution matches
one of your expressions, mark it. When you have a
BINGO shout it out!!!

PATTERNS AND EQUATIONS
WHAT COMES NEXT?
PATTERNS

What is a pattern? Create one.

What do you notice about this pattern?
MATHEMATICAL PATTERNS

3, 6, 9, 12, …

1, .5, .25, .125, …

100, 200, 300, 400, …

1, 4, 9, 16, 25, 36, …
CREATING EQUATIONS FOR A PATTERN

We need to write an equation to find a specific
term..

How could you do that?
EQUATIONS FOR MATHEMATICAL PATTERNS

3, 6, 9, 12, …

1, .5, .25, .125, …

100, 200, 300, 400, …

1, 4, 9, 16, 25, 36, …
MATHEMATICAL PATTERNS ACTIVITY

In your groups, create at least 5 challenging
patterns. DO NOT write the answers to the
patterns on the paper!

When you are done, hand your patterns into
me.

Each group will get another groups patterns to
create equations for.
VERBAL MODELS

What are some terms used for each of these
mathematical expressions?
 Addition?
 Subtraction?
 Multiplication?
 Division?
 Equals?
EQUATIONS AND INEQUALITIES
DEFINITIONS

What is the difference between an equation
and an inequality?
 Equation:
 Inequality:

Give three real-life examples of each of these
inequalities in your group. Be prepared to share
them.
PERCENTAGES/DISCOUNTS

Grab a laptop and go to this website:
 http://www.mathplayground.com/mathatthemall2.
html
Read through the ‘How To’ section, then
continue to the game.
 When you have received all four gold coins
bring your computer to the front and I will check
it.

VARIABLES AND TRANSLATION
Defining the variables means assigning letters
to an unknown quantity in the problem
 Translating the sentence means that you
change the word expression into a
mathematical expression containing variables
and mathematical operations with an equal
sign or an inequality sign

VARIABLES AND TRANSLATION

Key Words:
TRANSLATE VERBAL MODELS

Define the variable and translate the
expression for each example:
A
number plus 12 is 20
 9 less than twice a number is 33
 5 more than four times a number is 21
 $20 was one quarter of the money spent on the
pizza.
FINDING SOLUTIONS TO VERBAL MODELS
Bob worked for 2 hours and packed 24 boxes.
How much time did he spend on packing one
box?
 After a 20% discount, a book costs $12. How
much was the book before the discount?

 Remember
money off!
a discount means that you are taking
CHECK SOLUTIONS

Check that x = 5 is a solution to the equation
3x + 2 = -2x + 27

Check that z = 25 is a solution to the inequality
z + 1 < z -20
GROUP ACTIVITY

In a group of three, create three verbal models
like previous examples.

Put your group members names on them and
put them on the front desk when completed.

You will receive another groups questions at
the end of class to finish before you leave.
REAL-WORLD PROBLEMS

Tomatoes cost $0.50 each and avocados cost
$2.00 each. Anne buys six more tomatoes than
avocados. Her total bill is $8.00. How many
tomatoes and how many avocados did Anne
buy?
 Set
this up and solve within your groups.
REAL-WORLD PROBLEMS

In groups of two you are going to create your
own real-world problem much like the one we
just completed.
 Create
the story problem or situation (get creative)
on one of your notecards.
 Work out the answer on the other note card.
 Make sure your name is on both notecards for full
credit.
 Hand both notecards to me when finished.
FUNCTIONS AS RULES AND TABLES
WHAT IS A FUNCTION?

Function: A relation between a set of inputs and
a set of permissible outputs with the property
that each input is related to exactly one output.
 Each
input has exactly one output.
RULES AND TABLES
GRAPHS

To test to determine if a graph on a coordinate
plane is a function you want to test using the
vertical line test.
 Vertical
line test: If you run a vertical line across the
length of your coordinate plane every point on that
line should only touch the function at one point. If it
ever touches at two points, then the graph is not a
function.
IS IT A FUNCTION? (ORDERED PAIRS)

{(-5,-2), (-1,1), (3, -6), (8,1)}

{(2, -9), (2, -2), (6,8), (8,1), (11,-7)}
IS IT A FUNCTION? (TABLE)
IS IT A FUNCTION? (GRAPHS)
FUNCTIONS AS GRAPHS
GRAPHS
Coordinate Plane (AKA Cartesian Plane):
coordinate grid with a horizontal (x) and vertical
(y) number lines.
 Coordinate points are
written as (x, y).
 Origin is (0,0).

COORDINATE PLANE QUADRANTS
MINI- ACTIVITY
On the floor there is a coordinate plane.
 In your groups we are going to play a game.

 Each
group chooses one group member. That group
member is going to race to their box, grab a random
coordinate and race to that corresponding
coordinate on the floor.
 For each correct placing, groups receive one point.
For each incorrect placing, groups loose one point.
 Group with the most points receive candy.
WHAT IS A FUNCTION?

Group work:
 Search
function. What is it? Does it have any
properties?
 What
do domain and range mean?
GRAPHING FUNCTIONS

Given the table of values, graph the function.
What is the domain?
 What is the range?

x
-2
-1
0
1
2
y
6
8
10
12
14
DIFFERENT TYPES OF FUNCTIONS

In your small group you are going to look up the
properties of the graphs of your assigned type
of graph– use old textbooks in the back
cabinet, preferably Algebra 1 and 2 books.
 Linear
Graph
 Exponential Graph
 Quadratic Graph
 Absolute Value Graph
 Square Root Graph
TYPES OF GRAPHS

Linear

Exponential

Absolute Value

Square Root
DETERMINE IF A RELATION IS A FUNCTION

For each input there is EXACTLY one output!!!
 Determine
 (1,3)
 (-3,
if the relation is a function:
(-1, -2) (3, 5) (2, 5) (3,4)
20) (-5, 25) (-1, 5) (7, 12) (9, 2)
VERTICAL LINE TEST

If you can draw a vertical (up and down) line
that crosses the graph in more than one place,
then the relation is not a function.

Determine why this works!?
USING A GRAPH
What
year was
the ppm
335?
 What
was the
CO2 in
1947?

PROBLEM-SOLVING PLAN
SOLVE THE FOLLOWING PROBLEM

Group work:
A
coffee maker is on sale at 50% off the regular
ticket price. On the “Sunday Super Sale” the same
coffee maker is on sale at an additional 40% off. If
the final price is $21.00, what was the original
price of the coffee maker?

Share your solution and work with the class.
STRATEGIES

How did you solve this problem?

What different strategies did you use to solve
this problem?

Did everyone solve this the same way? Is one
way better than another way?
PROBLEM SOLVING PLAN
1. Read and understand the problem
 2. Make a plan to solve the problem
 3. Solve the problem
 4. Check the results

READ AND UNDERSTAND THE PROBLEM

Answer the following questions:
 What
am I trying to find out?
 What information have I been given?
 Have I solved a problem similar to this before?
 Define your known and unknowns
MAKE A PLAN TO SOLVE THE PROBLEM

Common strategies
 Draw
a diagram
 Make a table
 Look for a pattern
 Guess and check
 Work backwards
 Use a formula
 Write equations
SOLVE THE PROBLEM

Use your plan to solve the problem.
CHECK THE RESULTS
ALWAYS check your results to make sure they
make sense.
 Plug the answer back in and make sure
everything works

PROBLEM SOLVING STRATEGIES WORKSHEET

Individually work on the problem solving
worksheet / mini-activity